Let Ω be a convex domain with smooth boundary in Rd. It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on Ω is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane...

Introduction Compressed sensing (CS) has been shown to provide accurate reconstructions from highly undersampled data for certain types of MR acquisitions [1, 2]. This offers the promise of faster MR acquisitions, and further speed gains are possible when CS is used in conjunction with parallel acquisition schemes such as SENSE [3]. Several approaches have been recently proposed to reconstruct ...

In this paper, we prove that every F ∗ space (i.e., Hausdorff topological vector space satisfying the first countable axiom) can be characterized by means of its “standard generating family of pseudo-norms”. By using the standard generating family of pseudo-norms P, the concepts of P-bounded set and γ-maxpseudo-norm-subadditive operator in F ∗ space are introduced. The uniform boundedness princ...

Bose and Chaudhuri (1960) have introduced a class of binary errorcorrecting codes of block length 2 ~ 1, m = 2,3,• • which we refer to here as B-C codes. An efficient decoding scheme has been devised for them by Peterson (1960), and generalized to the natural extension of the B-C Codes to codes in pm symbols by D. Gorenstein and N. Zierler (1960). Two further properties of B-C codes are establi...

We make a probabilistic analysis related to some inference rules which play an important role in nonmonotonic reasoning. In a coherence-based setting, we study the extensions of a probability assessment defined on n conditional events to their quasi conjunction, and by exploiting duality, to their quasi disjunction. The lower and upper bounds coincide with some well known t-norms and t-conorms:...

We devise variants of classical nonconforming methods for symmetric elliptic problems. These variants differ from the original ones only by transforming discrete test functions into conforming functions before applying the load functional. We derive and discuss conditions on these transformations implying that the ensuing method is quasi-optimal and that its quasioptimality constant coincides w...

This paper follows the recent discussion on the sparse solution recovery with quasi-norms lq, q ∈ (0, 1) when the sensing matrix possesses a Restricted Isometry Constant δ2k (RIC). Our key tool is an improvement on a version of “the converse of a generalized Cauchy-Schwarz inequality” extended to the setting of quasi-norm. We show that, if δ2k ≤ 1/2, any minimizer of the lq minimization, at lea...

Let Xn = {x j }j=1 be a set of n points in the d-cube Id := [0, 1]d , and n = {φ j }j=1 a family of n functions on Id . We consider the approximate recovery of functions f on Id from the sampled values f (x1), . . . , f (xn), by the linear sampling algorithm Ln(Xn, n, f ) := ∑nj=1 f (x j )φ j . The error of sampling recovery is measured in the norm of the space Lq(I)-norm or the energy quasi-no...

Suppose that we observe entries or, more generally, linear combinations of entries of an unknown m×T -matrix A corrupted by noise. We are particularly interested in the high-dimensional setting where the numbermT of unknown entries can be much larger than the sample size N . Motivated by several applications, we consider estimation of matrix A under the assumption that it has small rank. This c...

This paper follows the recent discussion on the sparse solution recovery with quasi-norms `q, q ∈ (0, 1) when the sensing matrix possesses a Restricted Isometry Constant δ2k (RIC). Our key tool is an improvement on a version of “the converse of a generalized Cauchy-Schwarz inequality” extended to the setting of quasi-norm. We show that, if δ2k ≤ 1/2, any minimizer of the lq minimization, at lea...